Abstract
We called graph G non-singular if adjacency matrix A (G) of G is non-singular. A connected graph with n vertices and n-1, n and n+1 edges are called the tree, the unicyclic graph and the bicyclic graph. Respectively, as we all know, each connected bicyclic graph must contain ∞(a,s,b) or θ(p,l,q) as the induced subgraph. In this paper, by using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained.
Highlights
This paper considers only finite undirected simple graphs
By using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained
Let G be a graph with order n, the matrix A is defined as the adjacency matrix of graph G, which is de
Summary
This paper considers only finite undirected simple graphs. Let G be a graph with order n, the matrix A is defined as the adjacency matrix of graph G, which is de-. The number of nonzero eigenvalues and zero eigenvalues in the spectrum of the graph G are called the rant and nullity of the graph G, and denoted by r (G) and η (G) , respectively, obviously r (G) +η (G) = n. Sinogowitz put forward a question that how to find out all singular graphs (equivalently, to show all the nonsingular graphs), namely, to describe all graphs with nullity are greater than zero, it was a very difficult problem only with some special results [3] [4] [5] [6] [7]. By using three graph transformations which do not change the singularity, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained. The notation and terminology that are not described here are provided in [1]
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